The trilinear embedding theorem

Abstract

Let $σ_i$, $i=1,2,3$, denote positive Borel measures on $\mathbb{R}^n$, let $\mathcal{D}$ denote the usual collection of dyadic cubes in $\mathbb{R}^n$ and let $K:\,\mathcal{D}\to[0,\infty)$ be a map. In this paper we give a characterization of the trilinear embedding theorem. That is, we give a characterization of the inequality $$ \sum_{Q\in\mathcal{D}} K(Q)\prod_{i=1}^3\left|\int_{Q}f_i\,dσ_i\right| \le C \prod_{i=1}^3 \|f_i\|_{L^{p_i}(dσ_i)} $$ in terms of discrete Wolff's potential and Sawyer's checking condition, when $1<p_1,p_2,p_3<\infty$ and $\frac1{p_1}+\frac1{p_2}+\frac1{p_3}\ge 1$.